Role of Exact Exchange and Relativistic Approximations in Calculating

Article pubs.acs.org/JCTC

Role of Exact Exchange and Relativistic Approximations in Calculating 19F Magnetic Shielding in Solids Using a Cluster Ansatz Fahri Alkan,† Sean T. Holmes,‡ and Cecil Dybowski*,§ †

Department of Chemistry, Kansas State University, Manhattan, Kansas 66506, United States Department of Chemistry and Biochemistry, University of Windsor, Windsor, ON N9B 3P4, Canada § Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716, United States ‡

S Supporting Information *

ABSTRACT: Calculations of 19F magnetic shielding in various materials are presented. In calculations on gas-phase molecules, the variation of magnetic shielding with the amount of Hartree−Fock exchange (HFX) in the functional demonstrates that excellent agreement with experiment is obtained with an admixture of 50%, here denoted PBE0 (50%). Calculations at the PBE, PBE0 (25%), and PBE0 (50%) levels on 10 crystalline organofluorines and 15 crystalline inorganic fluorides, in which a cluster ansatz is used to model the lattice environment, were performed. For fluorine-containing aromatics, increasing the admixture of HFX results in the prediction of larger magnetic-shielding spans, whereas increasing the admixture of HFX in calculations for CFCl3 decreases the span. In calculations of 19F magnetic shielding of the inorganic fluorides, the use of sufficiently large clusters of inorganic fluorides results in accuracies similar to those calculated for the organofluorines. Relativistic effects on the magnetic shielding of inorganic fluorides, modeled with ZORA at both the scalar and spin−orbit levels, are dominated by the scalar terms that increase the shielding of most 19F sites over the nonrelativistic results. These effects appear to scale with the atomic number of the cation. For most elements of the sixth row (Cs, Ba, La, and Pb), the scalar relativistic contribution to the magnetic shielding is in the range of 20−77 ppm. For elements of group XII (Zn, Cd, and Hg) bonded to fluorine, the scalar relativistic contribution results in deshielding of the 19F site. and second-period nuclides such as 13C, 15N, 29Si, and 31P have been presented for a variety of model chemistries, and it is typically possible to obtain excellent agreement between structure-based theoretical calculations and experimental results, provided a suitable model chemistry is employed.24−27 Unfortunately, conventional computational approaches based on density-functional theory (DFT) are often incapable of quantitative prediction of 19F magnetic shielding to the degree necessary for assignment of spectra that allows assessment of proposed structures for unknown materials.25,28−35 Benchmark studies of 19F magnetic shielding of gas-phase molecules by Harding et al. illustrate that large deviations of DFT-computed magnetic-shielding parameters from experimental or advanced ab initio parameters are frequently observed.36 For the case of the F2 molecule, Schreckenbach and Ziegler have reported that standard DFT approximations underestimate the energy of the HOMO−LUMO (π → σ*) transition, resulting in substantial overestimation of the paramagnetic contribution to the magnetic shielding.37 The magnitudes of such errors in computed 19F magnetic shieldings are often system-dependent. Fukaya and Ono reported that computed magnetic shieldings of fluorine sites in oxygen-, nitrogen-, and sulfur-containing

1. INTRODUCTION The diverse chemistry of fluorine results from its ability to form bonds with nearly all elements, including the noble gases, and its ability to exist in a variety of co-ordination environments.1 The bonding motifs of fluorine reflect its high electronegativity, a characteristic that strongly influences properties that depend on electronic state, such as ionization potential and shielding in nuclear magnetic resonance (NMR) spectra. Because of the direct connection between electronic state and NMR parameters, 19F NMR spectroscopy has been used to analyze fluorine-containing materials, particularly solids, since the early days of the technique. The characteristic NMR properties of the 19F nucleus, including the large magnetogyric ratio, its 100% natural isotopic abundance, and the fact that it is a spin-1/2 particle,2−7 make the technique a sensitive analytical method for addressing questions about fluorine-containing materials. Whether the application is to small organic molecules,8−12 pharmaceuticals,13−15 proteins,16,17 cell membranes,18 polymers,19−21 superionic conductors,22 or ceramics,23 NMR parameters such as the 19F magnetic shielding reveal unusual electronic features of the fluorine environment. The prediction of magnetic-shielding tensors from structure by computational chemistry provides the link between electronic structure and experimental NMR parameters. Benchmark calculations of magnetic-shielding tensors of first© XXXX American Chemical Society

Received: May 30, 2017 Published: September 20, 2017 A

DOI: 10.1021/acs.jctc.7b00555 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation

Table 1. Experimental 19F Isotropic Chemical Shifts and Magnetic Shieldings of Gas-Phase Molecules, Computed Magnetic Shieldings for Eight Model Chemistries, and Statistical Data Associated with the Correlation between Calculated Magnetic Shielding and Experimental Chemical Shifta,b PBE

PBE0 (15%)

PBE0 (25%)

PBE0 (35%)

PBE0 (50%)

B3LYP

BHandHLYP

HF

molecule

δiso (ppm)

experimental σiso (ppm)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

CH3F HF SiF4 CF2H2 BF3 CF3H CF4 F2CO CFCl3 NF3 σref (ppm) m R2

−274.9 −213.9 −167.5 −143 −131.5 −78 −62.9 −25.5 0.0 145.8

463.6 402.6 356.2 331.7 320.2 266.7 251.6 214.2 188.7 42.9

457.5 409.4 335.2 309.6 296.9 240.3 225.1 177.9 134.4 −5.4 150.5 −1.132 0.996

461.8 408.8 341.3 320.5 304.7 251.6 234.1 187.2 156.5 13.1 164.6 −1.087 0.998

464.5 408.6 345.1 327.2 309.8 258.7 240.0 193.2 169.9 25.0 173.5 −1.059 0.998

467.2 408.6 348.8 333.5 314.7 265.6 245.7 199.1 182.6 36.5 182.1 −1.032 0.998

471.1 408.8 354.2 342.5 321.7 275.4 254.0 207.7 200.2 53.2 194.4 −0.994 0.998

460.9 407.7 343.9 320.9 307.5 252.6 234.8 189.9 162.3 9.0 165.4 −1.089 0.999

469.3 407.8 354.6 338.5 321.5 271.8 251.3 207.7 197.6 42.3 189.9 −1.012 0.999

478.7 407.6 371.7 361.7 343.3 299.4 277.3 235.0 244.5 94.2 226.4 −0.893 0.992

a Magnetic-shielding constants are reported relative on a scale on which the magnetic shielding of CFCl3 is assigned the value of 188.7 ppm. Chemical shifts are reported relative to CFCl3, which is assigned a chemical shift of 0.0 ppm. bExperimental data taken from refs 62 and 63.

molecules are often less reliable than 19F magnetic shielding of fluorine sites in fluorocarbons.38 It is common for DFT methods to overestimate the components of the 19F chemical-shift tensor derived from models of the solid structure by as much as 15− 40%.25,28,30,34,39−42 Systematic overestimation of shielding tensors of other first- and second-period nuclides is typically no more than 5−10%,24−26,43,44 and the selection of a suitable model chemistry removes much of the systematic deviation for calculations involving these nuclei, resulting in random errors of 1−3% of the chemical-shift range for nuclei like 13C.24 Further studies of 19F are required if calculations of the magneticshielding parameters of this important nucleus are to be used successfully to solve structural problems. The majority of computational studies of 19F magneticshielding parameters in solids have been performed using the gauge-including projector-augmented-wave (GIPAW) procedure of Pickard and Mauri, which models the extended lattice structure with periodic-boundary conditions, expands the valence wave function in a basis of plane waves, and replaces core−valence interactions with pseudopotentials.45,46 Although successful in many cases, GIPAW is limited to calculations based on pure DFT methods, specifically the local-density approximation (LDA) or the generalized-gradient approximation (GGA) in many plane wave DFT codes.47 Sadoc et al. have explored empirical correction of the GGA PBE functional for calculating 19F magnetic shielding in metal fluorides.28 In that study, the local potentials of Ca2+ (3d0), Sc3+ (3d0), or La3+ (4d0) ultrasoft pseudopotentials were artificially shifted to higher energies. This procedure decreases the covalent character of the metal−fluorine bond by reducing the interaction between the unoccupied metal d states and occupied fluorine p states, bringing calculated 19F chemical shifts into significant agreement with experiment. Laskowski et al. have applied the augmented-plane-waveplus-local-orbital (APW+lo) technique to the calculation of 19F magnetic shielding, which allows magnetic shielding to be computed using hybrid DFT approaches that include an admixture of Hartree−Fock exchange (HFX).39 For the particular set of metal fluorides considered in their study, the

hybrid DFT technique overcorrects the band gap and results in an underestimation of 19F chemical shifts.39 Cluster-based techniques for modeling magnetic-shielding tensors of many nuclei in crystalline solids have, in some cases, yielded accuracies that match or exceed those commonly obtainable by calculations in a periodic framework.24−26,48−52 One such study has benchmarked GGA, meta-GGA, and hybrid DFT methods for calculating 19F magnetic-shielding tensors of organic solids.25 The results of that study demonstrate that calculations employing meta-GGA and hybrid functionals improve significantly upon the results generated by using GGA functionals.25 In this article, we report a computational investigation of 19F magnetic-shielding tensors in a variety of bonding environments, including both molecular and network solids. The results illustrate that excellent agreement with experimental 19F chemical-shift tensors is obtained by altering the admixture of HFX in the PBE0 functional. The role of relativistic effects, including spin−orbit coupling, on computed 19F magnetic shielding is assessed for systems containing heavy atoms.

2. COMPUTATIONAL DETAILS 2.1. Benchmark Calculations of 19F Shielding in GasPhase Molecules. All calculations were performed using the Amsterdam Density Functional (ADF2016) suite of programs.53−55 Magnetic-shielding tensors were calculated within the gauge-including atomic orbital (GIAO) formalism,56,57 as implemented in ADF.58−61 Isotropic 19F magnetic-shielding constants were calculated for 10 gas-phase molecules [CH3F, HF, SiF4, CH2F2, BF3, CF3H, CF4, F2CO, CFCl3, and NF3], which span an isotropic 19 F chemical-shift range of 421 ppm.62,63 Magnetic-shielding parameters calculated for these systems at the CCSD(T) level are known to agree well with experimental values36 and are used here to assess several DFT functionals for the calculation of 19F magnetic shielding. Calculations for these systems were performed using geometries obtained at the B3LYP/TZ2P level.64,65 Table 1 presents experimental isotropic chemical shifts, experimental magnetic shieldings (obtained using an absoluteshielding scale on which the isotropic shielding of CFCl3 is B


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Figure 1. Dependence of the linear-regression parameters, (a) σref and (b) m, on the percentage of Hartree−Fock exchange (HFX) incorporated into various functionals used in the calculation. Horizontal lines indicate either the experimental isotropic reference shielding of 188.7 ppm (for σref) or the ideal slope of −1.00 (for m).

188.7 ppm, as suggested by Jameson et al.),62,63 and computed magnetic-shielding parameters obtained using several model chemistries. Values were computed with the standard GGA functional of Perdew, Burke, and Ernzerhof (PBE);66 with hybrid versions of this functional using 15%, 25% (commonly referred to as PBE0),67 35%, and 50% HFX; with Becke’s threeparameter exchange functional and Lee−Yang−Parr correlation (B3LYP);64,65 with Becke’s half-and-half exchange functional and LYP correlation (BHandHLYP);65 and with HF theory. Table 1 also presents statistical data associated with the correlation between the calculated magnetic shieldings and experimental chemical shifts, including the slopes (m), extrapolated shieldings of the reference compound (σref), and linear-correlation coefficients (R2). The results in Table 1 illustrate the strong linear relationship between calculated 19F magnetic shielding and experimental 19F chemicals shift. The relationship between the experimental isotropic chemical shifts, δiso,j, and the calculated isotropic magnetic shielding, σiso,j, in Table 1 is modeled by eq 1 δiso, j = σref + mσiso, j

optimizations, as obtained in previous work.25 The materials in this study were 1,3,5-trifluorobenzene,9,73 perfluorobenzene,8,74 perfluoronaphthalene,10,75 2-fluorobenzoic acid,9,76 4fluorobenzoic acid,9,77 2-fluorotoluene,9,78 3-fluorotoluene,9,78 4-fluorotoluene,9,78 p-fluoranil,9,79 and CFCl3.12,80 The 19F magnetic-shielding tensors of these materials were calculated on clusters of molecules, each of which represents a local portion of the extended lattice environment. The molecules selected for inclusion in the cluster were determined by the symmetryadapted-cluster (SAC) ansatz.24,25 The accuracy of this cluster approximation has been previously rigorously benchmarked, and its use is known to yield excellent agreement with calculations employing periodic-boundary conditions, such as those obtained using GIPAW.24,25 In all calculations, a basisset-partitioning scheme was used, in which the central molecule of the cluster (containing the NMR-active 19F sites) was given the large basis set TZ2P and all other molecules were given the smaller basis set DZ. Two example clusters, illustrative of the basis-set-partitioning scheme, are shown in Figure 2. Calculations were performed using the DFT methods PBE, PBE0 (25%), and PBE0 (50%).

(1)

Slopes, m, for the correlation lines vary between −0.893 and −1.132, and R2 is greater than 0.992 in all cases. The relationship between the linear-regression parameters m and σref and the proportion of HFX is shown in Figure 1. Of significance is the trend in computed results as the percentage of HFX in the functional is increased, particularly in the series beginning with PBE and ending with PBE0 (50%). As 50% HFX is approached, both m and σref approach their values in the ideal case (−1.00 and 188.7 ppm, respectively). Results obtained at the BHandHLYP level agree very well with those obtained at the PBE0 (50%) level. The worst results are obtained at the HF and PBE levels. The critical role of HFX in DFT calculations of NMR parameters, including magnetic shielding and spin−spin coupling constants, is well established.68,69 Several studies have examined the effects of HFX on the calculation of spin−spin coupling constants between 19F and other nuclei.70−72 For example, Garciá de la Vega and San Fabián report that calculations of 2JFF and 3JFF spin−spin coupling constants generally agree better with experiment when the admixture of HFX is increased beyond the percentage in most standard DFT functionals. 2.2. Calculations of 19F Shielding in Organic Molecular Solids. 19F magnetic-shielding tensors for 10 organic molecular solids with 12 unique NMR-active sites are reported. Calculations were carried out on structures derived from single-crystal X-ray diffraction studies, with further refinements of atomic positions from plane-wave DFT geometry

Figure 2. Two example clusters, (a) perfluorobenzene and (b) perfluoronaphthalene, including a schematic of the partitioning of the basis sets.

2.3. Calculations of 19F Shielding in Network Solids. F magnetic-shielding tensors for 15 inorganic fluorides were computed using the cluster ansatz. The clusters included up to the second coordination shell (CS) of atoms around the central 19 F nucleus, as shown in Figure 3. The materials in this study were LiF,28,81 NaF,28,82 MgF2,2,83 AlF3,84,85 KF,28,86 CaF2,28,87 ZnF2,2,83 RbF,28,88 SrF2,28,89 CdF2,90,91 CsF,28,92 BaF2,28,93 LaF3,28,94 HgF2,90,95 and PbF2.96,97 Clusters were terminated by employing valence modification of terminal atoms using bond valence theory98−100 (VMTA/BV).26,49−51 The all-electron TZ2P basis set was employed for the whole cluster. In our analysis of density functionals in Section 3.2, the ZORA Hamiltonian at the scalar level was employed for the systems 19

C


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Table 2. Statistical Data for the Relationships between Calculated Principal Components of 19F Magnetic-Shielding Tensors and Principal Components of 19F Chemical-Shift Tensors for 12 Fluorine Sites in Organic Molecular Solids by Various Models

Figure 3. Second-coordination-shell clusters for (a) NaF and (b) CaF2. Terminal atoms treated with VMTA/BV model are indicated by a light yellow halo around the atom.

model chemistry

PBE/ TZ2P

PBE0 (25%)/ TZ2P

PBE0 (50%)/ TZ2P

PW91/ GIPAWa

PW91/ ccpVTZa

PW91/ pcSseg3a

σref (ppm) m R2

129 −1.15 0.984

161 −1.07 0.988

189 −1.00 0.988

115 −1.20 0.987

132 −1.16 0.986

127 −1.17 0.986

a

These values are based on a re-analysis of a subset of the results in ref 25.

RbF, SrF2, CdF2, CsF, BaF2, LaF3, HgF2, and PbF2. In Section 3.3, the ZORA Hamiltonian101,102 was employed at both the scalar and spin−orbit levels for comparison of different relativistic effects on the magnetic shielding of 19F nuclei.

for several nuclei.26,51 GIPAW calculations for 29Si sites yield a reference 34 ppm less shielded than the reference shielding predicted by clusters; this difference is 61 ppm for 31P.26 For the 119Sn reference shielding, the relationship is more complicated, with the size of the difference varying significantly among tin sites of different oxidation state.51 For the 19F magnetic-shielding calculations reported here, the differences between GIPAW and cluster calculations are due to the errors introduced by cluster modeling (vide inf ra), differences in the basis sets employed with these methods, and possibly errors introduced from the use of pseudopotentials in the GIPAW calculations. For gas-phase molecules, comparison between the calculated magnetic shieldings with all-electron basis sets and GIPAW shows that the GIPAW calculations systematically underestimate the calculated 19F magnetic-shieldings compared to the calculations employing the all-electron basis set.28 Cluster-based calculation of 19F magnetic shielding in organofluorines using the Gaussian-type basis sets cc-pVTZ and pcSseg-3 yields similar results to the all-electron calculations reported here using Slater-type basis sets (Table 2).25 These differences should be taken into account when comparing the performance of the cluster model and the GIPAW method. The principal components of predicted magnetic-shielding tensors for the sites in the 10 organic solids are reported in Table 3, along with the experimental values. Experimental magnetic-shielding constants are reported on an absolute scale, assuming that the isotropic shielding of CFCl3 (δ = 0.0 ppm) is 188.7 ppm.62,63 An important feature of this analysis is the variation of the magnetic-shielding span (Ω = |σ11 − σ33|) as a function of the proportion of HFX in the functional. The effect

3. RESULTS AND DISCUSSION 3.1. Organic Solids. Figure 4 illustrates the relationship between the principal components of calculated 19F magneticshielding tensors and the principal components of experimental 19 F chemical-shift tensors for 10 fluorine-containing organic solids. Calculations were performed with the PBE functional (Figure 2a), the PBE0 (25%) functional (Figure 2b), and the PBE0 (50%) functional (Figure 2c). In Table 2 are statistical data associated with the correlation between calculated and experimental values derived from the organic solids. The calculated 19F magnetic-shielding tensor components become progressively more shielded as the percentage of HFX in the functional is increased. Such a systematic change can be seen in the extrapolated reference shielding as a function of the amount of HFX included. As the percentage of HFX in the calculation is increased, the HOMO−LUMO gap increases, leading to a reduction in magnitude of the negative paramagnetic contribution to the shielding. The slope of the correlation line also varies systematically with the amount of HFX included. For calculations at the PBE0 (50%) level, the values of the reference shielding and slope do not differ significantly from the ideal case (188.7 ppm and −1.00, respectively).62,63 We have previously discussed differences in extrapolated reference shieldings between GIPAW and cluster calculations

Figure 4. Relationship between the principal components of calculated 19F magnetic-shielding tensors and the principal components of experimental 19 F chemical-shift tensors. Calculated values were obtained using (a) the PBE functional, (b) the PBE0 (25%) functional, and (c) the PBE0 (50%) functional. D


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Table 3. Experimental and Calculated Principal Components of 19F Magnetic-Shielding Tensors, Isotropic Magnetic Shielding, Span, and Residual between Calculated and Experimental Principal Values of Magnetic-Shielding Tensors for 12 Fluorine Sites in Organic Molecular Solids

a

σ11

σ22

σ33

σiso

Ω

residual

material

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)a

CFCl3 PBE PBE0 (25%) PBE0 (50%) 2-toluene PBE PBE0 (25%) PBE0 (50%) 3-toluene PBE PBE0 (25%) PBE0 (50%) 4-toluene PBE PBE0 (25%) PBE0 (50%) p-fluoranil PBE PBE0 (25%) PBE0 (50%) 1,3,5-trifluorobenzene PBE PBE0 (25%) PBE0 (50%) 3-fluorobenzoic acid PBE PBE0 (25%) PBE0 (50%) 4-fluorobenzoic acid PBE PBE0 (25%) PBE0 (50%) perfluorobenzene PBE PBE0 (25%) PBE0 (50%) perfluoronaphthalene (F1) PBE PBE0 (25%) PBE0 (50%) perfluoronaphthalene (F2) PBE PBE0 (25%) PBE0 (50%) perfluoronaphthalene (F3) PBE PBE0 (25%) PBE0 (50%)

155 97 126 153 249 193 219 243 250 178 203 225 251 184 209 234 258 178 215 242 233 175 199 221 242 186 209 229 262 207 232 256 309 250 277 299 291 241 264 282 287 231 253 272 286 226 244 260

155 103 133 160 337 287 314 334 330 269 298 319 312 250 280 304 309 232 261 288 314 265 293 314 325 280 305 324 338 295 320 340 309 269 286 303 291 247 271 293 287 244 271 294 286 231 258 283

278 194 235 267 369 348 364 378 360 325 348 362 387 353 366 380 459 438 446 453 375 337 351 365 367 327 344 359 368 343 357 371 467 440 449 456 456 434 442 450 464 433 440 448 465 435 441 448

196 131 165 193 318 276 318 299 313 258 283 302 317 262 285 306 342 283 307 328 307 259 281 300 311 265 286 304 322 281 303 322 361 320 337 352 346 307 326 342 346 303 321 338 345 297 315 330

123 98 109 114 120 155 146 135 110 147 145 137 136 169 157 146 201 260 231 211 -142 161 153 145 125 141 135 130 106 136 125 115 158 191 172 157 165 193 178 167 177 202 188 175 179 209 197 187

Residual =

1 3

66 32 7 45 22 6 58 48 16 56 33 11 65 38 15 49 27 9 47 26 9 43 21 4 44 25 9 40 21 6 44 25 13 50 32 18

3

∑t = 1 (σiicalc − σiiexp)2

of the proportion of HFX on computed 19F magnetic shielding depends on the co-ordination environment of the 19F site. In aromatic compounds, the span decreases with increasing proportion of HFX, with the best agreement with experimental values obtained at the PBE0 (50%) level for all 11 sites. However, for CFCl3, the only example of a nonaromatic molecular solid studied here, the span increases with increasing

proportion of HFX, with the best agreement with experiment also obtained at the PBE0 (50%) level. Residuals between calculated and experimental 19F magnetic-shielding tensors for all fluorine sites are highest at the PBE level and lowest at the PBE0 (50%) level (Table 3). By agreement with the generally accepted values of the reference shielding and by the E


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Figure 5. Correlation between the calculated 19F magnetic shielding and experimental chemical shift. Calculated values were obtained using (a) the PBE functional, (b) the PBE0 (25%) functional, and (c) the PBE0 (50%) functional.

minimization of the residuals, the best agreement with experiment is obtained by calculation at the PBE0 (50%) level. The variation of the 19F span with the admixture of HFX is related to the hybridization state (sp3 or sp2) of the directly bonded carbon atom. A similar effect has been observed for calculated 13C magnetic-shielding tensors.25 In the latter case, modeling the relationship between calculated magnetic shieldings and experimental chemical shifts reveals that sp3and sp2-hybridized 13C sites belong to statistically distinct subpopulations, when calculated using a pure DFT functional. This effect is notably absent when a hybrid DFT functional is employed. To expand on the present investigation of 19F chemical shifts, we calculated 19F spans for several fluorinecontaining gases [CF4, CH3F, CH2F2, CFCl3, CHF3, CFCl3, SiF4, and C6H6 (Table S1)]. The variation of Ω with the admixture of HFX is negligible in the cases of CF4, CH3F, and CHF3. The span of CFCl3 increases in the gas-phase molecule, whereas the span of C6F6 decreases, reproducing the trends observed in the solids. For CH2F2 and SiF4, increasing the HFX admixture decreases Ω. In the former case (CH2F2), the molecular point group and the orientation of the principal components of the 19F magnetic-shielding tensor are different than observed for the other molecules, whereas for the latter case (SiF4), the fluorine atom is bonded to silicon rather than carbon. The hybridization of the carbon atom, the resulting molecular symmetry, and the orientation of the magnetic shielding principal component all affect the observed trends between Ω and the HFX admixture. 3.2. Inorganic Solids. Figure 5 presents correlations between calculated magnetic shieldings and experimental 19F chemical shifts at the PBE, PBE0 (25%), and PBE0 (50%) levels. For MgF2 and ZnF2, the three principal components of the 19F chemical-shift tensor are presented, as they are experimentally available.2 For the remaining systems, only isotropic values are considered in the correlations. Each 19F site is given equal statistical weight, independent of the type of experimental data available. Table 4 gives the parameters of the linear correlations for this set of inorganic solids. The correlation plots and linear-regression parameters for all systems investigated (organic and inorganic) are available in the Supporting Information (Figure S1 and Table S2). For the inorganic solids, the slope of the linear regression with PBE-based calculations shows a large deviation (around 40%) from the ideal value of −1.00. The predicted absolute shielding for the reference compound is 106 ppm, which is considerably different from the experimental absolute shielding

Table 4. Parameters of the Best-Fit Linear Correlation of Calculated Magnetic Shielding versus Experimental Chemical Shift for the 19F Nuclei in Inorganic Solids model chemistry

PBE

PBE0 (25%)

PBE0 (50%)

GIPAW/PBEa

σref (ppm) m R2

106 −1.39 0.967

164 −1.18 0.985

204 −1.06 0.985

86 −1.38 0.965

a

These data are taken from ref 28.

(188.7 ppm).62,63 For a similar set of inorganic fluorides, Sadoc et al. calculated magnetic shieldings of 19F using the GIPAW method with the PBE functional.28 In that study, they found a linear regression slope of −1.38, very close to the slope obtained in this study (−1.39) using cluster models. The predicted shielding of the reference compound determined from their GIPAW calculation is 20 ppm less shielded than that found here with model clusters (Table 4), consistent with our observations for the organofluorines. Similar to the results for organofluorines, the slope and the extrapolated reference shielding improve significantly when using the PBE0 (25%) functional compared to PBE. There is also improvement of the scatter around the best-fit line. At the PBE0 (50%) level, the slope of the linear regression is only 6% different from that for the ideal case. For each system, increasing the amount of HFX in the functional produces more shielded values. The increase in the computed magnetic shielding is system-dependent. For example, the difference between PBE and PBE0 (50%) for σiso in MgF2 is only 16 ppm, whereas the equivalent differences for the three fluorine sites in LaF3 range between 112 and 125 ppm. The large variation in magnetic shielding with the admixture of HFX is most likely due to the change in the calculated HOMO−LUMO gap. For MgF2, the increase in the HOMO−LUMO gap between PBE and PBE0 (50%) calculations is 6.3 eV, whereas the equivalent increase is 8.1−8.6 eV for LaF3. However, interpreting variation in calculated 19F magnetic shielding in terms of HOMO−LUMO gaps alone can be misleading. In the case of CaF2, the difference in σiso between PBE and PBE0 (50%) calculations is 72 ppm despite the change in the calculated HOMO−LUMO gap being similar to the difference observed for MgF2 (16 ppm). The large change in σiso for CaF2, compared to MgF2, can be rationalized by the observation that the LUMO level for CaF2 originates from Ca atomic d states with the PBE functional, whereas with PBE0 (50%), the LUMO mostly originates from Ca atomic s states. Similar effects in the electronic configuration with the inclusion F


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Journal of Chemical Theory and Computation of HFX are observed for other group II fluorides. The exception to this observation is MgF2, which does not have access to a low-lying metal d state. Consequently, the shielding of this 19F site does not vary greatly with the admixture of HFX. In general, the effect of HFX on the 19F magnetic shielding becomes larger with increasing atomic number of the cation in the fluorine-containing inorganic compound. The effect of HFX on the calculated magnetic shielding for the inorganic systems is tabulated in the Supporting Information (Table S3). At the PBE0 (50%) level, the predicted reference shielding obtained from the linear-regression parameters (Table 4) of inorganic fluorides is 15 ppm more shielded than the experimental value, and it is more shielded than the predicted reference shielding obtained from the linear-regression parameters of the organofluorines (Table 2). In Table 5, we

functionals and large fourth-CS clusters remain feasible only for a handful of systems. 3.3. Relativistic Effects on 19F Magnetic Shielding. Inclusion of relativistic effects is necessary for accurate predictions of magnetic-shielding tensors for systems containing heavy atoms.48−51 For 19F nuclei, one expects relativistic effects on the magnetic shielding through the influence of the so-called heavy-atom-on-light-atom (HALA) mechanism.103,104 We investigate the importance of HALA effects on the accuracy of calculated 19F magnetic shielding for several selected inorganic fluorides (LiF, NaF, CaF2, ZnF2, RbF, SrF2, CdF2, CsF, BaF2, LaF3, HgF2, and PbF2). Calculations are carried out with the PBE0 (50%) functional. To account for relativistic effects, the ZORA Hamiltonian is employed at either the scalar or the spin−orbit level. In Figure 6a, the correlation between calculated 19F isotropic magnetic shielding and experimental isotropic chemical shift is shown for calculations at the non-relativistic DFT level of theory. Figure 6b,c shows similar correlations at the ZORA/ scalar DFT and ZORA/spin−orbit DFT levels of theory, respectively. Table 6 presents linear-regression parameters for

Table 5. Comparison of Calculated Principal Components of 19 F Magnetic-Shielding Tensors and Isotropic Magnetic Shielding Using Second-Coordination-Shell and FourthCoordination-Shell Clusters for MgF2, AlF3, and ZnF2a material MgF2 second-CS cluster fourth-CS cluster AlF3 second-CS cluster fourth-CS cluster ZnF2 second-CS cluster fourth-CS cluster a

σ11

σ22

σ33

σiso

σiso (exp.)

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

382 369

408 393

429 410

406 391

382

337 338

348 340

425 409

370 362

362

398 390

428 407

445 429

424 409

391

Table 6. Parameters of the Best-Fit Linear Correlation of Calculated Isotropic Magnetic Shielding with Experimental Chemical Shift for Inorganic Fluorides at Different Levels of Inclusion of Relativistic Effects model chemistry

non-relativistic

ZORA/scalar

ZORA/spin−orbit

σref (ppm) m R2

162 −1.31 0.942

204 −1.06 0.982

205 −1.06 0.987

these different treatments of relativistic effects. At the nonrelativistic DFT level, the slope of the best-fit line deviates by 31% from the ideal case. There is also substantial scatter around the best-fit line. When scalar relativistic effects are included at the ZORA level, the agreement between theory and experiment improves relative to the non-relativistic case. The slope is within 6% of the ideal case, and the scatter about the best-fit line is diminished, as seen from the value of R2. There is little change in the linear-regression parameters relative to the scalar level when the spin−orbit effects are included in the ZORA Hamiltonian to predict the isotropic shielding (Figure 6c). A more detailed examination of relativistic effects on 19F nuclei in crystalline inorganic fluorides is found in Table 7, where isotropic magnetic shieldings at the non-relativistic DFT

All calculations are performed with the PBE0 (50%) functional.

report the comparison of principal components and the isotropic values of the magnetic-shielding tensors calculated using the second- and fourth-CS clusters of selected inorganic fluorides. For these systems, the isotropic magnetic shieldings calculated with fourth-CS clusters are deshielded by 8−15 ppm relative to the values obtained from the calculations with second-CS clusters. The agreement between experimental and calculated isotropic magnetic shieldings also improves when fourth-CS clusters are used. These results demonstrate that it is possible to improve the performance of calculations with fourth-CS clusters. However, such calculations involving hybrid

Figure 6. Effect of level of inclusion of effects of relativity on correlation between the calculated 19F magnetic isotropic shielding and experimental isotropic chemical shift. Calculated values were obtained at the (a) non-relativistic PBE0 (50%), (b) ZORA/scalar/PBE0 (50%), and (c) ZORA/ spin−orbit/PBE0 (50%) levels. G


Article

Journal of Chemical Theory and Computation Table 7. Calculated 19F Isotropic Magnetic Shielding Computed at the Non-relativistic PBE0 (50%) Level, as Well as Scalar and Spin−Orbit Corrections to the Shielding σiso (non-relativistic)

ΔσSC

ΔσSC+SO

material

(ppm)

(ppm)

(ppm)a

F− LiF NaF CaF2 ZnF2 RbF SrF2 CdF2 CsF BaF2 LaF3, F1 LaF3, F2 LaF3, F3 HgF2 PbF2, F1 PbF2, F2

481 411 432 310 424 297 294 421 219 214 193 118 138 443 136 176

0 0 0 1 −3 7 7 −6 22 26 34 47 46 −23 71 72

3 2 3 4 (3) −2 (+1) 8 (+1) 8 (+1) −6 (0) 20 (−2) 25 (−1) 36 (+2) 46 (−1) 44 (−2) −30 (−7) 77 (+6) 77 (+5)

shown that the accuracies of magnetic-shielding predictions are affected mostly by the inclusion of the spin−orbit terms in the ZORA Hamiltonian. Furthermore, spin−orbit contributions (through the HALA mechanism) are important for light nuclei such as 1H, 13C, or 29Si when heavy nuclei are also present in the material.103−112 Our results in Tables 6 and 7 indicate that the accuracies of predicted 19F magnetic shieldings in inorganic fluorides are mostly determined by the inclusion of scalar effects in the ZORA Hamiltonian. The relationship between spin−orbit effects and magnetic shieldings has been associated with the s-character of the heavy atom−light atom bond113−115 and with the covalence of the metal−ligand bond.106,111,116 To understand better the small effect of spin−orbit coupling on 19F magnetic shielding found in the present study, we performed magnetic-shielding calculations for the light atom in the Pb(XHy)2 series (X = F, O, N, C, and y = 0, 1, 2, 3, respectively). The calculated spin− orbit-only contributions are 15, 22, 58, and 73 ppm for 19F, 17 O, 15N, and 13C nuclei, respectively, exhibiting a strong correlation between the covalent character of metal−ligand bond and the magnitude of the spin−orbit effects. These results demonstrate that the small spin−orbit contributions observed for the calculated 19F magnetic shieldings are most likely due to the large ionicity, and the resulting small s-type character, in the metal−fluorine bond. Figure 7 shows the relationship between the absolute value of the relativistic contribution to the 19F isotropic magnetic

a

The spin−orbit only contributions to F19 magnetic shieldings are shown in parentheses.

level are given, along with relativistic corrections at the scalar and spin−orbit levels of approximation. ΔσSC is the difference between the calculated isotropic magnetic shielding at the ZORA/scalar level and the isotropic magnetic shielding calculated at the non-relativistic level. ΔσSC+SO is the difference between the isotropic magnetic shielding calculated at the ZORA/spin−orbit level (which also includes scalar effects) and the isotropic magnetic shielding at the non-relativistic level. For a free fluoride ion, the scalar correction to the magnetic shielding is negligible, and the spin−orbit correction is 3 ppm (Table 7). In principle, the spin−orbit contributions should vanish due to the spherical symmetry of the F− anion. For this reason, the calculated spin−orbit contribution to the fluoride ion should be treated as the numerical uncertainty for the discussion on spin−orbit contributions to the 19F magnetic shielding in the other systems. The relativistic corrections are small for LiF, NaF, CaF2, and ZnF2, indicating that relativistic effects due to the presence of the cation are minimal for these inorganic fluorides. For RbF, SrF2 and CdF2, the relativistic corrections to the magnetic shieldings are noticeable; however, they are less than 10 ppm in all cases. For the inorganic fluorides in which the fluorine is coordinated to sixth-row metals (CsF, BaF2, LaF3, HgF2, and PbF2), the relativistic corrections become important for the accuracy of the predicted magnetic shieldings of 19F nuclei, as the differences range between 20 and 77 ppm. In most cases, inclusion of relativistic effects causes the 19F nuclei to be more shielded, except when the fluorine is coordinated to the group XII elements (Zn, Cd, and Hg), where the relativistic contributions to the magnetic shielding are negative (vide inf ra). An interesting result from the data in Table 7 is this: relativistic corrections to the 19F isotropic magnetic shieldings are mostly dominated by scalar effects. Spin−orbit contributions are only a few ppm for most inorganic fluorides. As expected, the largest spin−orbit contributions are seen for HgF2 and PbF2, with spin−orbit corrections above the scalar correction ranging between 5 and 7 ppm. For heavier nuclei such as 207Pb, 199Hg, and 119Sn,48−51 previous studies have

Figure 7. Contribution of relativistic effects (SC + SO) to isotropic magnetic shielding of 19F-containing solids as a function of the atomic number of the cation (M) in MFn (n = 1, 2, 3). HgF2, CdF2, and ZnF2 are shown as blue points.

shielding as a function of the atomic number of the cation to which the fluorine atom is bonded, for the systems shown in Table 7. In general, the absolute values of the relativistic contributions increase with the increasing atomic number. However, there are some exceptions. The relativistic contributions to the magnetic shielding for SrF2 and CdF2 are 8 and 6 ppm, respectively, indicating a decrease in relativistic contribution, while the atomic number increases by 10. For HgF2 and PbF2, there is a 47 ppm difference in the absolute relativistic contributions to the magnetic shielding, or a 107 ppm difference if one considers the different signs of the relativistic contributions, although the difference between the atomic numbers of Pb and Hg atoms is only 2. A similar, but smaller, deviation from this trend is observed for ZnF2. For ZnF2, CdF2, and HgF2, the contribution of relativistic effects is negative (Table 7), indicating that relativistic H


Article

Journal of Chemical Theory and Computation contributions to 19F magnetic shielding are influenced by factors other than the atomic number of the cation. The LUMO levels of the group XII fluorides originate mainly from the metal s states, which undergo s orbital contraction with the inclusion of scalar effects. As a result, the HOMO−LUMO gaps decrease for the group XII fluorides with the inclusion of scalar effects, which probably affects the magnitude and sign of relativistic contributions to the calculated 19F isotropic magnetic shielding compared to the other systems. Previous work on heavy nuclei has shown that inclusion of relativistic effects is crucial for accurate predictions of the principal components of magnetic-shielding tensors.48−51 To understand the contribution of HALA effects on the principal components and span (Ω) of the 19F magnetic shielding tensor, Table 8 gives calculated principal components of PbF2 and

the accurate prediction of 19F magnetic shielding. For all systems investigated (organic and inorganic), the best agreement between theory and experiment is achieved when 50% HFX is employed in the density functional. For all systems, the accuracy of the predicted magnetic shielding using cluster models and the PBE functional is quite similar to that obtained with the GIPAW formalism.28,34 In comparison, using model clusters and the PBE0 (50%) density functional provides agreement between experiment and theory for organofluorines similar to the agreement achieved for 13Ccontaining solids.24 For inorganic fluorides, the agreement with experiment at the PBE0 (50%) is also substantially improved, relative to results obtained at the PBE level. The agreement with experiment for the inorganic fluorides is not as good as that obtained for calculations on organofluorines. This difference between the two sets is due, in part, to the size of the clusters employed to model the inorganic fluorides. Inclusion of the contribution of relativistic HALA effects on the magnetic shielding improves agreement between experiment and theory, particularly in the case of sixth-row metal fluorides. As expected, the higher the atomic number of the heavy atom, the more significant are contributions of relativistic effects on the magnetic shielding of fluorine atoms bound to a heavy atom. Relativistic effects on fluorine magnetic shielding are dominated by scalar contributions, whereas the inclusion of the spin−orbit terms changes the calculated magnetic shielding by only about 7 ppm, at most.

Table 8. Comparison of Calculated Principal Components of 19 F Magnetic-Shielding Tensors and Spans of 19F Nuclei in LaF3 and PbF2 at the Non-relativistic, Scalar Relativistic, and Spin−Orbit Relativistic Levels of Theorya material LaF3, site I non-relativistic scalar spin−orbit LaF3, site II non-relativistic scalar spin−orbit LaF3, site III non-relativistic scalar spin−orbit PbF2, site I non-relativistic scalar spin−orbit PbF2, site II non-relativistic scalar spin−orbit a

σ11

σ22

σ33

Ω

(ppm)

(ppm)

(ppm)

(ppm)

107 153 153

180 216 218

292 311 315

185 158 162

10 72 69

172 212 212

172 213 213

162 141 144

30 89 85

192 231 230

192 232 231

163 143 146

118 193 197

122 197 202

168 233 240

51 40 42

126 214 219

188 256 261

214 273 279

87 58 60

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00555. Chemical-shift spans of gas-phase molecules; statistical data associated with the prediction of all 19F magneticshielding tensors, independent of subclass; and effect of HFX on the calculated isotropic 19F magnetic shielding of network solids (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

All calculations are performed with PBE0 (50%) functional.

Cecil Dybowski: 0000-0002-0557-8915 Funding

C.D. acknowledges the support of the National Science Foundation under Grants CHE-0956006 and DMR-1608366.

LaF3. There are large differences between principal components calculated at the non-relativistic level and those calculated at the scalar relativistic level. These differences range between 19 and 92 ppm. The scalar relativistic effects on the calculated principal components are not uniform, and the largest scalar relativistic contributions in each system are observed for the σ 11 component. Calculated spans decrease for each system when scalar effects are included. Similar to our findings for the isotropic magnetic shielding, spin−orbit contributions to the calculated principal components and spans of the 19F nuclei are also quite small, ranging from 0 to 7 ppm.

Notes

The authors declare no competing financial interest.

■

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